Diffusions with Rough Drifts and Stochastic Symplectic Maps

Wednesday, March 4, 2015 -
3:00pm to 4:00pm
This is a joint seminar with Analysis of Fluids and Related Topics.  According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $d/r+2/q<1$. As an application one show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $d/r+2/q<1$. As our second application we derive a Constantin-Iyer type circulation formula for certain weak solutions of Navier-Stokes equation.
Speaker: 
Fraydoun Rezakhanlou
UC Berkeley
Event Location: 
Fine Hall 322